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Tunable Zero-index Photonic Crystal Waveguide for Two-qubit Entanglement Detection Ege Özgün, Ekmel Ozbay, and Humeyra Caglayan ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00576 • Publication Date (Web): 17 Oct 2016 Downloaded from http://pubs.acs.org on October 20, 2016
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Tunable Zero-index Photonic Crystal Waveguide for Two-qubit Entanglement Detection Ege Özgün,∗ † Ekmel Ozbay,† ‡ ¶ and Humeyra Caglayan∗ † ,
, ,
,
E-mail:
[email protected];
[email protected] Abstract
Quantum entanglement is one of the most counterintuitive outcomes of quantum mechanics and is central in quantum information science. 1 It is the leading actor in various phenomena such as quantum teleportation, 2,3 quantum cryptography 4 and quantum computation. 5 It is essential to quantify the entanglement, for a systematic study of the quantum information platforms. There exists dierent strategies for de ning the degree of quantum entanglement. Concurrence 6 is one of the most commonly used entanglement measures, which we will also utilise through this manuscript. There is a tremendous eort to devise a method for consistently preparing and measuring entanglement. 7,8 An experimental work in 2010 utilises a superconducting circuit to prepare and measure three-qubit entanglement. 9 The motivation behind obtaining a multi-qubit entangled state is its necessity for quantum error correction schemes, which starts with tripartite entanglement. The realisation of multiqubit entanglement was previously achieved for spins, 10 ions 11 and photons. 12 There also exists numerous studies utilising WGs to achieve entanglement. 1316 Despite these eorts, a more robust and accessible strategy is still necessary for another step forward in quantum information, since although the above mentioned strategies are successful, it would be dramatically di cult to integrate them into a quantum information device. Quantum dots (QDs) play a signi cant role in quantum computation 17 and can be integrated into WGs, which are quite accessible. Therefore, to overcome these
Preparation and measurement of an entangled state are essential in quantum information applications. The recent theoretical works on preparing and measuring entangled states via coupled quantum dot-waveguide platforms are promising. However, an alternative strategy is still needed for more accessible implementations. In this letter, we propose a simple but powerful medium for quantum information processing and communication consisting of a zeroindex photonic crystal (PC) waveguide (WG). By exploiting the zero-index behaviour of the PC WG, we overcome the di culty of sensitivity in quantum dots' location, which is one of the main challenges for preparing and measuring a two-qubit entangled state. Moreover, our platform possesses the feature of tunability, which allows it to operate in dierent wavelength regimes. The suggested medium is also promising for realising long range spatial coherence as well as long range entanglement.
Keywords Quantum entanglement, photonic crystals, zero-index waveguide To whom correspondence should be addressed Nanotechnology Research Center, Bilkent University, 06800 Ankara, Turkey ‡ Department of Physics, Bilkent University, 06800 Ankara, Turkey ¶ Department of Electrical and Electronics Engineering, Bilkent University, 06800 Ankara, Turkey ∗ †
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Figure 1: (a) The dimensionless frequency a/λ versus the dimensionless mode index ka/2π is plotted, depicting the disallowed regions, photonic band gaps and the guiding mode. (b) Top view of the WG-QD coupled system with an illustration of signi cant parameters. quency which yields a zero mode index for the WG. These materials are referred as ENZ materials. 22 Among dierent ways of realising such materials, 23 the most straightforward one is exploiting the dispersion of the WG by operating near its cut-o frequency. 24,25 It is important to note that although our platform eectively operates within the ENZ region, we are not working in the ultra-strong coupling regime, in which even the rotating wave approximation can breakdown. 26 The ENZ medium stretches the eective wavelength inside the structure, and as a result the distance (d) between the sources and/or scatterers, although physically sizeable, becomes electrically small compared with the effective wavelength (λef f ). In other words, here d/λef f becomes small, not because d is small, but because λef f becomes very large due to the eect of the ENZ medium. As a result, ENZ materials oer several exciting potential applications, such as: supercoupling and tunnelling, 22 photon density of states enhancement, 27 directive emission, 28 enhanced
problems, a WG-QD combined platform is a very promising candidate. There are two recent proposals that uses WG-QD coupled schemes. The rst one utilises a plasmonic WG operated near its cut-o frequency to attain epsilon nearzero (ENZ) behaviour 18 whereas the other one uses the correspondence between two-photon detection probability and concurrence which is a manifestation of the symmetry obtained by forcing the QDs to interact at the origin. 19 Although those are powerful strategies, a tunable and more accessible alternative would be of high interest for robust quantum information applications. There are also two recent proposals for controlled coupling of photon emitters: The rst one is a nanowire-based PC WG system, which shows a modi ed Mollow triplet spectrum 20 and the second is a three-dimensional polariton WG structure. 21 We propose a platform based on the above mentioned two-photon detection probability scheme and enhance it by using a tunable platform of PC WG operated near its cut-o fre-
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Figure 2: Magnitudes and phases of the E- eld for the InP PC WG for dierent WG widths (w) are shown. From top to bottom, the zero-index behaviour is obtained for the wavelengths 1.55140, 1.54297 and 1.48341 μm respectively. The w = 0.7488 μm case corresponds to removing a single line of rods. By changing w, we can tune the wavelength for which zero-index behaviour is observed. ter position insensitive. Here, we propose an all-dielectric PC WG as an ENZ medium for quantum information applications, for the rst time. The suggested platform is accessible both in terms of its fabrication and the excitation of the quantum emitters inside the WG. PCs are ideal platforms for achieving con nement and guided transmission of electromagnetic waves. Moreover, they give rise to numerous interesting phenomena including negative refraction, 33 slow waves 34 and nonreciprocal chirality and asymmetric transmission. 35 In this letter, we suggest an InP based PC WG as a platform for preparing and measuring a two-
transmission 29 and radiation phase front shaping. 30,31 Instead of having cavities that enhance the electric eld (E- eld) in a sub-wavelength area with limited eective mode areas, 32 the suggested nanophotonic devices will eectively stretch the eective mode to the whole structure. In homogeneous electromagnetic media, the wavelength de nes the spatial scale and a very large wavelength implies a very low frequency. However in an ENZ medium, the WG mode has an extremely extended wavelength; while its frequency lies within an optical or telecom band. Therefore, ENZ medium creates an environment, where the interaction is emit-
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where c†k /ck are bosonic creation/annihilation operators, Vk(j) 's are the photon-QD interactions, k is the wave vector of the photon, (j) (j) σ+ /σ− are the raising/lowering operators for the QDs, Γj 's contains the dissipation eects, including the Ohmic losses and losses to the WG. We will exploit the ENZ nature of the platform, hence we can place the QDs with any separation as long as k 1/d is satis ed. Since we are operating our WG near its cut-o frequency, we can have su ciently small values for k , which gives us the freedom in our choice of d. Therefore we can safely assume that the QDs are interacting at the origin and yet at the same time they are distant enough so that we can neglect the direct interaction between them. In other words, mathematically, there is no difference between placing the QDs at the origin or placing them with any separation d, unless k 1/d is violated. Zero-index behaviour of the WG not only simpli es the algebra, but also is essential for the measurement strategy which is available for a certain symmetry con guration, that is obtained when the QDs are interacting at the origin. 19 For our system, the concurrence is given by:
qubit entangled state. To obtain the suggested PC WG, a line defect is opened in the middle of the square lattice of cylindrical InP rods with radii r = 0.2a, a being the lattice constant of the PC. The fabrication procedure of the InP rods are described in Ref. 36. The band structure of the InP PC WG is given in Fig. 1a and relevant parameters are sketched in Fig. 1b. Fig. 2. illustrates the zero-index regimes obtained for dierent WG widths (w). Left panels show the magnitude of the E- eld whereas the constant phases shown in the right panels are the clear signatures of the zero-index behaviour. By varying w, it is possible to obtain zero-index platforms operating in dierent wavelengths, which is one of the signi cant advantages of our proposal. The w=0.8010 μm case gives a zero-index WG for λ = 1.55140 μm, which is quite favourable, since most of the detectors work in the vicinity of 1.55 μm. The spectral bandwidth of the ENZ regime is approximately ∼ 11 nm which can be seen from the second gure given in the Supporting Information. The other important component of the suggested platform is QDs. The incoming WG mode ωk must be large enough to excite the QDs meaning ωk > ωj , where ωj 's, with j = (1, 2) denote the transition frequencies of the QDs. Moreover, QDs absorption frequencies must coincide with the cut-o frequency of the WG. Thanks to the zero-index behaviour of the PC WG, the positions of the QDs are quite exible, which is essential for the measurement procedure.
Γ2 + 4Δ212 Γ2 + 4 C = 2g12 2 2 Γ + 4Δ212 + g12 Γ2 + 4
We will treat the QDs as two-level systems, begin with a Jaynes-Cummings type Hamiltonian for the coupled WG-QD system 3740 in the rotating wave approximation and take h ¯ = 1: k
+
ωk c†k ck
+
2
(j) Vk
+
(j) ck σ +
j=1
Γj (j) (j) σ+ σ− ωj − i 2 j=1
2
(j) c†k σ−
(2)
where the dimensionless parameters are de ned as: Δ12 ≡ Δ1 /Δ2 , g12 ≡ g1 /g2 and Γ ≡ Γ1 /Δ2 with Δ1 ≡ − ω1 and Δ2 ≡ − ω2 , = ωk is the eigenenergy of the system and g1 and g2 are the single-mode WG-QD interactions. Here equal losses from the QDs are assumed, i.e., Γ1 = Γ2 . Details of the derivation is given in the Supporting Information. Fig. 3 summarises the results of the analytical calculations for the concurrence. Concurrence is plotted against varying g12 and Δ12 values for dierent values of Γ in Fig. 3a. To focus on the condition ( g1 = g2 ) that is required for the measurement proposal, the g12 = 1 slice is shown in Fig. 3b.
The model
ˆ = H
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Figure 3: (a) Concurrence is shown for Γ = 0.01 (top left), 0.1 (top right), 1 (bottom left), 2 (bottom right) versus g12 = g1 /g2 and Δ12 = Δ1 /Δ2 . (b) Slices from the left panel are plotted for g12 = 1, which is required to obtain the correspondence between two-photon total detection probability and concurrence, that the measurement strategy is based on.
Measurement strategy
in Ref. 19. The advantage of our platform within the context of the measurement strategy is two-fold: Firstly it enables separation insensitivity for the QDs, which makes the measurement strategy realisable, and secondly thanks to the tunable nature of the PC WG, one can choose to work in the desired wavelength, which is crucial for the choice of QDs and the detectors to be used.
By de ning odd and even operators, assuming equal photon-waveguide interaction strengths (g1 = g2 ) and recasting the Hamiltonian, one sees that the interaction term contains only even operators. By exploiting this result, the concurrence can be related to a physically measurable entity, 19 namely two-photon total detection probability (PRR ),
P (ξ)
2Re[ξ]
RR − 1 = =C
PRR (ξ = 0)
1 + |ξ|2
Design
(3)
Before concluding, we will suggest a realisable design with the currently available QDs. Thanks to the tunable nature of our platform, we have a great freedom in our choice of QDs and the PC WG to be used. Since most detectors operate at 1550 nm, it would be reasonable to choose our QDs and the PC WG such that they also operate in that telecom wavelength regime. Hence, the InP based PC WG we mentioned, which is achievable in terms of fabrication, is a good candidate for the PC WG, since it can be tuned to yield zero-index behaviour at the wavelength of 1550 nm. There exist com-
where ξ = α/β with α and β being the probability amplitudes for QD1 and QD2 to absorb a photon and make a transition from their ground to excited states and PRR (ξ = 0) is a fully disentangled state. Two separate measurements are required to extract the concurrence information as Eq. 3 illustrates: The rst one is needed to measure PRR (ξ = 0), whereas the second one is for the two-photon total detection probability for the entangled state, i.e. PRR (ξ). The details of the measurement procedure is described
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mercial QDs whose emission/absorption spectra are also within that range. So by exploiting the tunability of the platform we suggested, one can achieve a design operating at the telecom wavelength.
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In conclusion, we have suggested a novel platform for quantum entanglement measurement, which is both tunable and accessible. Our platform eliminates the di culty in QD placement by exploiting the ENZ behaviour. Moreover, the suggested PC WG structure is also accessible in terms of fabrication and also has the signi cant feature of tunability. Finally the suggested platform with additional QDs, is a strong candidate for quantum error correction schemes. Therefore, the suggested zero-index PC WG platform can open up new horizons in quantum information science.
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Acknowledgement Authors would like to thank Hodjat Hajian and Mirbek Turduev for their help with the simulations and Ceyhun Bulutay for his comments on the manuscript. This work is supported by the projects DPTHAMIT and NATO-SET-193. One of the authors (E.O.) also acknowledges partial support from the Turkish Academy of Sciences.
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